🎛️Control Theory Unit 3 – Time–domain analysis and design

Key Concepts and Definitions

• Time-domain analysis examines system behavior over time, focusing on transient and steady-state responses
• Transient response refers to the system's behavior during the initial period after an input is applied
• Steady-state response describes the system's behavior after the transient response has settled
• Stability in the time domain implies that the system's output remains bounded for bounded inputs
  • Asymptotic stability means the system's output converges to a specific value as time approaches infinity
  • Marginal stability indicates the system's output oscillates with a constant amplitude
• Steady-state error quantifies the difference between the desired and actual outputs in the steady state
• Rise time measures the time required for the system's output to rise from 10% to 90% of its final value
• Settling time is the time taken for the system's output to settle within a specified percentage of its final value (typically 2% or 5%)
• Overshoot represents the maximum deviation of the system's output above its final value, expressed as a percentage

Time-Domain Representation of Systems

• Time-domain representation of systems involves differential equations that describe the relationship between input and output signals
• Linear time-invariant (LTI) systems are commonly represented using transfer functions in the Laplace domain
• The transfer function $G(s)$ is the ratio of the Laplace transform of the output $Y(s)$ to the Laplace transform of the input $U(s)$, assuming zero initial conditions
  • $G(s) = \frac{Y(s)}{U(s)}$
• The inverse Laplace transform of the transfer function yields the impulse response $g(t)$, which characterizes the system's response to a unit impulse input
• The convolution integral relates the input $u(t)$ and the impulse response $g(t)$ to the output $y(t)$
  • $y(t) = \int_{0}^{t} g(\tau)u(t-\tau)d\tau$
• State-space representation is an alternative time-domain representation that uses a set of first-order differential equations to describe the system dynamics
  • State variables capture the system's internal behavior and memory
  • State equations relate the state variables, inputs, and outputs

Transient Response Analysis

• Transient response analysis examines the system's behavior during the initial period after an input is applied
• The transient response is typically characterized by the system's response to standard test inputs, such as step, impulse, or ramp functions
• Step response is the system's output when subjected to a unit step input, which is a sudden change in the input from zero to one
  • The step response provides insights into the system's rise time, settling time, and overshoot
• Impulse response is the system's output when subjected to a unit impulse input, which is an infinitely short pulse with an area of one
  • The impulse response is the derivative of the step response and characterizes the system's dynamic behavior
• Ramp response is the system's output when subjected to a ramp input, which is a linearly increasing function of time
  • The ramp response helps assess the system's ability to track a constantly changing input
• Transient response can be analyzed using time-domain techniques, such as solving differential equations or using the convolution integral

Steady-State Error Analysis

• Steady-state error analysis evaluates the difference between the desired and actual outputs of a system in the steady state
• The steady-state error $e_{ss}$ is the limit of the error signal $e(t)$ as time approaches infinity
  • $e_{ss} = \lim_{t \to \infty} e(t)$
• For a unity feedback system with a forward path transfer function $G(s)$, the steady-state error depends on the type of input and the system type
  • System type refers to the number of pure integrators (poles at the origin) in the forward path transfer function
• For a step input, the steady-state error is given by $e_{ss} = \frac{1}{1 + \lim_{s \to 0} G(s)}$
  • A system with type 0 (no integrators) will have a non-zero steady-state error for a step input
  • A system with type 1 or higher will have zero steady-state error for a step input
• For a ramp input, the steady-state error is given by $e_{ss} = \frac{1}{\lim_{s \to 0} sG(s)}$
  • A system with type 1 (one integrator) will have a non-zero steady-state error for a ramp input
  • A system with type 2 or higher will have zero steady-state error for a ramp input
• Steady-state error can be reduced by increasing the system type or by using controllers with integral action

System Stability in Time Domain

• System stability in the time domain refers to the boundedness of the system's output for bounded inputs
• A stable system's output remains bounded and converges to a steady-state value when subjected to bounded inputs
• Asymptotic stability implies that the system's output converges to a specific value (usually zero) as time approaches infinity
  • For a linear system, asymptotic stability requires all poles of the transfer function to have negative real parts
• Marginal stability indicates that the system's output oscillates with a constant amplitude
  • Marginally stable systems have poles on the imaginary axis (zero real parts) and no poles with positive real parts
• Unstable systems have outputs that grow unbounded over time, even for bounded inputs
  • Unstable systems have at least one pole with a positive real part
• Time-domain stability can be assessed using various methods, such as the Routh-Hurwitz criterion or the root locus technique
• The Routh-Hurwitz criterion determines stability by examining the coefficients of the system's characteristic equation without explicitly solving for the roots
• The root locus technique graphically depicts the trajectories of the closed-loop system poles as a parameter (usually the gain) varies, allowing stability analysis and controller design

Performance Specifications and Design Criteria

• Performance specifications define the desired behavior of a control system in terms of time-domain characteristics
• Rise time ($t_r$) specifies the time required for the system's output to rise from 10% to 90% (or 0% to 100%) of its final value
  • Faster rise times indicate quicker system responses but may lead to increased overshoot and oscillations
• Settling time ($t_s$) is the time taken for the system's output to settle within a specified percentage (typically 2% or 5%) of its final value
  • Shorter settling times are generally desirable for improved system performance and reduced transient behavior
• Overshoot ($M_p$) represents the maximum deviation of the system's output above its final value, expressed as a percentage
  • Lower overshoot is preferred to minimize excessive output deviations and potential damage to the system
• Steady-state error ($e_{ss}$) quantifies the difference between the desired and actual outputs in the steady state
  • Minimizing steady-state error ensures accurate tracking of reference inputs and reduces long-term deviations
• Other performance specifications may include peak time, delay time, and maximum allowable control effort
• Design criteria are derived from the performance specifications and guide the controller design process
  • Examples include gain and phase margins, which quantify the system's robustness to uncertainties and variations
• Trade-offs often exist between different performance specifications, requiring a balance based on the specific application and requirements

Controller Design Techniques

• Controller design techniques aim to develop control strategies that meet the desired performance specifications and ensure system stability
• Proportional-Integral-Derivative (PID) control is a widely used technique that combines proportional, integral, and derivative actions
  • Proportional control provides an output proportional to the error signal, reducing the steady-state error but may lead to overshoot
  • Integral control eliminates steady-state error by accumulating the error over time but can introduce oscillations and instability
  • Derivative control improves transient response and damping but is sensitive to noise and high-frequency disturbances
• Lead compensation introduces a phase lead to improve the system's stability and transient response
  • Lead compensators increase the system's bandwidth and reduce the settling time but may amplify high-frequency noise
• Lag compensation introduces a phase lag to reduce steady-state error and improve low-frequency performance
  • Lag compensators attenuate high-frequency noise and disturbances but may slow down the system's response
• Pole placement is a state-space design technique that places the closed-loop system poles at desired locations to achieve specific performance characteristics
  • Pole placement requires full state feedback and may not be feasible for systems with limited sensor information
• Linear Quadratic Regulator (LQR) is an optimal control technique that minimizes a quadratic cost function involving the system states and control inputs
  • LQR provides a systematic approach to balancing performance and control effort but requires accurate system models and state measurements
• Robust control techniques, such as H-infinity control, address uncertainties and variations in the system model and ensure satisfactory performance under various conditions

Real-World Applications and Examples

• Automotive cruise control systems maintain a constant vehicle speed by adjusting the throttle based on the measured speed and desired set point
  • PID control is commonly used in cruise control systems to minimize speed deviations and ensure smooth operation
• Temperature control in industrial processes, such as chemical reactors or furnaces, relies on feedback control to maintain the desired temperature profile
  • Lead-lag compensation is often employed to handle the system's thermal dynamics and achieve precise temperature regulation
• Robotic manipulators require precise position and trajectory tracking control to perform tasks accurately and efficiently
  • Computed torque control, a form of feedback linearization, is used to decouple and linearize the manipulator dynamics for improved performance
• Active suspension systems in vehicles aim to enhance ride comfort and handling by actively controlling the suspension forces based on road conditions and driver inputs
  • LQR control is applied to optimize the trade-off between ride comfort and handling while considering actuator constraints
• Power system frequency regulation maintains the balance between power generation and demand to ensure stable grid operation
  • Proportional-Integral (PI) control is used to adjust the power output of generators in response to frequency deviations caused by load changes
• Autopilot systems in aircraft control the aircraft's attitude, altitude, and trajectory based on pilot inputs and navigation data
  • Robust control techniques are employed to handle uncertainties in the aircraft dynamics and ensure safe and reliable operation under various flight conditions